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Introduction to quantum field theory

A particle is moving under the influence of a 2 dimensional harmonic oscillator with the potential [math]V=\tfrac{m \omega^2}{2}(X^2 + Y^2)[/math].

- Write down the Hamiltonian in terms of the raising and lowering operators [math]\cal{H}=\cal{H} \left( \hat{a}_x,\hat{a}_x^{\dagger},\hat{a}_y,\hat{a}_y^{\dagger} \right)[/math]. What is the explicit form of all these operators in terms of the position and momentum operators?
- Using [math]\hat{a}_x \hat{a}_y\vert \Psi_{0,0}\rangle = 0[/math] find the normalized wave function [math]\vert\Psi_{0,0} (p_x, p_y)\rangle[/math]. Please pay attention to the momentum dependence of the wave function.
- Using the previous answers, calculate [math]\vert \Psi_{0,1}(p_x, p_y)\rangle [/math], [math]\vert \Psi_{1,0}(p_x, p_y)\rangle [/math], [math]\vert \Psi_{1,1}(p_x, p_y) \rangle [/math], and [math]\vert \Psi_{3,1}(p_x, p_y)\rangle[/math], where [math]\vert \Psi_{n_x,n_y}(p_x, p_y)\rangle[/math] is the wave function in momentum space while [math]n_{x}[/math] and [math]n_{y}[/math] label the energy levels.

We solved in class the Schrödinger equation for the spectrum of the hydrogen atom as if it were isolated in the Universe. In reality, H is in an environment, which means that the Coulomb potential does not extend to infinity, but gets screened by other charges. Using first-order perturbation theory, explore the simple case of a screened Coulomb potential cut-off at a constant distance R: [math]V(r)={-\kappa\over r}\theta(R-r)[/math], with [math]\kappa[/math] the square of the electron mass (in Gaussian units) and [math]\theta[/math] the Heaviside step function.

- How would you characterize the strength of the perturbation and under which conditions does perturbation theory give reliable results?
- How does the cutoff affect the spectrum? Does it lift degeneracies? Give specific answers for three values of [math]R[/math], one for gases under standard conditions, one for liquids and one for solids (and explain how you estimate those values). Is the approximation reliable in all three cases?
- Give detailed results for the energies and the wave functions in the [math]1s[/math] and [math]2s[/math] cases.

Two spin-½ particles of masses [math]m_1[/math] and [math]m_2[/math], interact via a potential that may be expressed as:

[math]V = \dfrac{-e^2}{r}\left(1 + \dfrac{b}{r}\sigma^{(1)}\cdot \sigma^{(2)} \right),[/math]

where [math]\sigma^{(i)}[/math] corresponds to the [math]i[/math]-th particle; [math]r = \lvert \vec{r}_{1} - \vec{r}_2 \rvert[/math] represents the magnitude of relative separation between the two particles; and [math]b[/math] is an arbitrary constant. Determine the bound-state energy spectrum of the combined two-particle system. What condition(s) do you need to impose upon [math]b[/math] to avoid unphysical eigenstates?